PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 279 



of the equations of relative motion. These integrals give, 



^?-^' = /3y'-y^' = B(p^-y;j), I (91.) 



and 



J(a^'-pa')+l(/3y'-y/3')+;?(ya'-«y')=:0; (92.) 



they contain therefore the known law of equable description of areas, and the law of 

 a plane relative orbit. If we take for simplicity this plane for the plane I v, the quan- 

 tities ^^'77' will vanish ; and we may put, 



I = r cos ^, ;? = r sin ^, ^ = 0, "^ 



> . . . (93.) 



oc = rQ cos 0Q, (6 = ?'o sm 0q, 7 = 0, J 



and 



^' = r' cos ^ — ^ r sin 0, vj = r' sin & ■{■ & r cos L ^ = 0, "I 



?■ . . . (94.) 



a' = r'o cos ^0 — ^'0 ^0 sin ^o> Z^' = ^o sin &q -\- &q r^ cos ^0, 7' = 0, J ' * ' 



the angles & &q being counted from some fixed line in the plane, and being such that 

 their difference 



^ - ^0 =^ ^ (95.) 



These values give 



IV — ;7|'=r2^, a|3' - ,5a' = ro2^'o, a;? - |3| = rroSin^, . . . .(96.) 



and therefore, by (88.) and (91), 



r2^ = ro2^o = ^; • W.) 



the quantity J ^ is therefore the constant areal velocity in the relative motion of the 

 system ; a result which is easily seen to be independent of the directions of the three 

 rectangular coordinates. The same values, (93.), (94.), give 



I cos ^ + ;? sin ^ = r, |' cos ^ + V sin ^ = r', a cos ^ -f j8 sin ^ = r^ cos ^, 



J(os., 



a COS ^0 + |3 sin 6q = r^, a' cos &q + 3' sin &q = r'o, I cos 6q-\- y} sin ^0 = ^ cos ^, ' ^ 

 and therefore, by the intermediate and final integrals, (89.), (90.), 



^' = ?, ''o = fo; (99.) 



results which evidently agree with the condition (T^.), and which give by (79.) and 

 (81.), for all directions of coordinates, 



r'2+ ^; -2(//^l^-77^2)/(r)= "] 



J> (100.) 



r^ + ^-2(^, + ^,)/(ro)=2H,(;i + l); J 



the other auxiliary quantity H^ is therefore also a constant, independent of the time, 



and enters as such into the constant part in the expression for \r''^ + -2 j the square 



of the relative velocity. The equation of condition (I^), connecting these two con- 



2 o 2 



